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We consider the problem of the construction of the estimator-process of the unknown finite-dimensional parameter in the case of the observations of nonlinear autoregressive process. The estimation is done in two or three steps. First we estimate the unknown parameter by a learning relatively short part of observations and then we use the one-step MLE idea to construct an-estimator process which is asymptotically equivalent to the MLE. To have the learning interval shorter we introduce the two-step procedure which leads to the asymptotically efficient estimator-process too. The presented results are illustrated with the help of two numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: We give a simple and entirely elementary proof of Gasper''s Theorem on the Markov sequence problem for Jacobi polynomials. It is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding molecules introduced by Marc Kac, and the methods developed here yield new estimates relevant to the collision model. [Copyright &y& Elsevier]

Two approaches to studying the correlation functions of the binary Markov sequences are considered. The first of them is based on the study of probability of occurring different "words" in the sequence. The other one uses recurrence relations for correlation functions. These methods are applied for two important particular classes of the Markov chains. These classes include the Markov chains with permutative conditional probability functions and the additive Markov chains with the small memory functions. The exciting property of the self-similarity (discovered in Phys. Rev. Lett.90, 110601 (2003) for the additive Markov chain with the step-wise memory function) is proved to be the intrinsic property of any permutative Markov chain. Applicability of the correlation functions of the additive Markov chains with the small memory functions to calculating the thermodynamic characteristics of the classical Ising spin chain with long-range interaction is discussed. [ABSTRACT FROM AUTHOR]

The paper consists of three parts. The first part is dedicated to a Markov monotonous random search on a general optimization space. Under certain restrictions, an upper bound for the complexity of search is presented in an integral form, suitable for further analysis. This estimate is applied to the case of a torus, where several specific results on the rate of convergence are obtained with the help of a supplementary optimization problem, discussed in Appendix. [Copyright &y& Elsevier]

A problem of an optimal stopping of a Markov sequence is considered. The existence conditions and the structure of optimal and ∊-optimal (∊ > 0) multiple stopping rules are obtained. [ABSTRACT FROM AUTHOR]